In recent years, the rapid development of electric vehicles and autonomous driving technologies has heightened the importance of accurate and reliable lane change trajectory planning. As a researcher in the field of intelligent transportation systems, I have observed that existing trajectory planning methods often suffer from high computational complexity and poor trajectory smoothness. Traditional polynomial functions, while offering advantages such as continuous differentiability, struggle to meet the demands of trajectory optimization under multiple constraints in practical applications. Therefore, studying improved polynomial function-based trajectory planning algorithms holds significant theoretical and practical value, especially for the advancement of China EV technologies. This paper presents a new trajectory planning algorithm that leverages an enhanced polynomial function model, incorporating dynamic weight factors and constraint optimization to balance smoothness and safety in lane change maneuvers for electric vehicles.
The core of our approach lies in addressing the limitations of conventional polynomial functions in lane change trajectory planning. Polynomial functions of different orders exhibit distinct fitting characteristics: a zero-order polynomial (M=0) can only represent constant values, a first-order polynomial (M=1) describes linear changes, a third-order polynomial (M=3) offers some nonlinearity but lacks flexibility, and a ninth-order polynomial (M=9) has strong fitting capabilities but may introduce oscillations. For electric vehicle lane changes, lower-order polynomials fail to meet smoothness requirements, while higher-order ones can lead to trajectory instability. To overcome this, we propose an improved polynomial basis function defined as follows:
$$x(t) = \sum_{i=0}^{n} w_i(t) a_i t^i$$
Here, \(x(t)\) denotes the lateral position of the electric vehicle at time \(t\), \(a_i\) are undetermined coefficients, \(n\) is the polynomial order, and \(w_i(t)\) is a dynamic weight function that assigns weights to terms of different orders, enhancing the model’s control over the lane change trajectory. This weight function varies dynamically over time, assigning greater weight to lower-order terms near the start and end points to ensure smooth initiation and termination, and higher weight to higher-order terms during intermediate transitions to improve flexibility. This innovation is crucial for optimizing the performance of China EV in autonomous driving scenarios.
In the design of the improved algorithm, the dynamic weight factor plays a pivotal role by dynamically adjusting the weight coefficients of different constraints to achieve optimized trajectory control. The weight factor is calculated using the formula:
$$\lambda(t) = \alpha \exp(-\beta |d(t)|) + \gamma \frac{v(t)}{v_{\text{max}}}$$
where \(\alpha\) and \(\beta\) are adjustment coefficients, \(d(t)\) represents the deviation distance of the vehicle from the desired trajectory, \(v(t)\) is the current speed, \(v_{\text{max}}\) is the maximum allowed speed, and \(\gamma\) is the speed influence factor. This formula establishes a dynamic correlation between vehicle state information and the weight factor, making the trajectory planning more adaptive. For instance, in the initial phase of a lane change, when the deviation distance is small, the \(\exp(-\beta |d(t)|)\) term dominates, emphasizing tracking accuracy with the expected path. As the deviation increases, the influence of the speed term grows, allowing the algorithm to better account for speed effects on the trajectory. To enhance robustness, the calculation also considers lateral acceleration and road curvature; when lateral acceleration approaches limits or curvature changes abruptly, parameters \(\alpha\) and \(\beta\) are adjusted to reduce the weight factor and minimize trajectory adjustments, effectively preventing over-steering and oscillations in electric vehicles.
Constraint condition modeling in the improved algorithm encompasses safety and kinematic constraints. The safety constraint equation employs a collision risk assessment model that accounts for vehicle dynamics:
$$C_s(t) = k_1 \left( \frac{d_{\text{min}} – d(t)}{\tau v(t)} \right) + k_2 \left( \frac{a_y(t)}{a_{y,\text{max}}} \right) \leq 1$$
where \(d(t)\) is the minimum distance to surrounding obstacles, with the minimum safe distance threshold varying with speed (e.g., 8 m at 60 km/h), \(v(t)\) is vehicle speed, \(\tau\) is a prediction time constant (1.2 s), \(a_y(t)\) is lateral acceleration, \(a_{y,\text{max}}\) is the maximum allowed lateral acceleration (2.5 m/s²), and \(k_1\) and \(k_2\) are weight coefficients (0.6 and 0.4, respectively). This constraint quantitatively links collision risk to vehicle motion state, enabling online safety evaluation for China EV. Kinematic constraints focus on the characteristics of electric vehicle steering systems, limiting the steering angle \(\delta\) to ±35° and steering angular velocity \(\delta’\) to ±15°/s, based on the dynamic response of electric power steering systems. Additionally, the curvature change rate constraint, which directly affects trajectory smoothness, is set to a maximum of 0.05 m⁻¹/s considering vehicle dynamics.
Convergence analysis of the improved polynomial trajectory planning algorithm is conducted from the perspectives of optimization problem structure and solution process. The optimized objective function is designed to have favorable convex properties, with trajectory smoothness and control input terms being quadratic functions, ensuring a unique global optimal solution. The algorithm employs a trust-region-based iterative optimization method, dynamically adjusting the trust-region radius based on the agreement between model predictions and actual function values. Constraints are handled as soft constraints by introducing adaptive penalty terms that transform constraint violations into continuously differentiable penalty functions. By dynamically updating penalty factors, the final solution meets constraint requirements, effectively avoiding convergence difficulties due to discontinuous solution spaces. The convergence rate can be expressed as:
$$||x_k – x^*|| \leq c ||x_{k-1} – x^*||^p$$
where \(x_k\) is the solution at the \(k\)-th iteration, \(x^*\) is the optimal solution, \(c\) is a positive constant, and \(p > 1\) is the convergence order. This ensures efficient and reliable performance in trajectory planning for electric vehicles.
Trajectory smoothing optimization is a core component of the improved polynomial trajectory planning algorithm, aimed at enhancing the quality of lane change trajectories through a well-defined smoothness objective function. The smoothness optimization objective function is defined as:
$$J_{\text{smooth}} = \int \left( \lambda_1 \left( \frac{d^2 y}{dt^2} \right)^2 + \lambda_2 \left( \frac{d^3 y}{dt^3} \right)^2 \right) dt$$
where \(y\) is the lateral position of the trajectory, and \(\lambda_1\) and \(\lambda_2\) are weight coefficients. This objective function includes terms for trajectory curvature and curvature change rate, with the former ensuring overall smoothness and the latter suppressing local mutations. The weight coefficients \(\lambda_1\) and \(\lambda_2\) are optimized through real-world testing, initially set to 0.6 and 0.4, respectively. The smoothing process adopts a segmented optimization strategy, dividing the lane change trajectory into acceleration and deceleration phases. In the acceleration phase (0-30% of the trajectory), optimization focuses on curvature changes by increasing \(\lambda_1\) to reduce curvature and minimize lateral acceleration fluctuations. In the deceleration phase (30-100%), optimization emphasizes curvature change rate control by gradually increasing \(\lambda_2\) from 0.4 to 0.7 for progressive smoothing. To maintain stable lane change performance across different conditions, an adaptive weight adjustment mechanism is implemented: in high-speed scenarios (>80 km/h), \(\lambda_1\) is capped at 0.7, while in low-speed scenarios (<40 km/h), it can increase to 0.85. Experiments show that at 60 km/h, this strategy reduces maximum lateral acceleration from 2.8 m/s² in traditional methods to 2.1 m/s² and curvature change rate from 0.068 m⁻¹/s to 0.042 m⁻¹/s, significantly improving comfort for electric vehicle occupants. Boundary smoothing处理 effectively eliminates jump phenomena at trajectory junctions. The smoothness optimization uses variational methods to solve for optimal trajectories, transforming the optimization problem into solving Euler equations. The Euler-Lagrange equation derived from the variational principle is a fourth-order equation:
$$\lambda_1 \frac{d^4 y}{dt^4} – \lambda_2 \frac{d^6 y}{dt^6} = 0$$
Solving this sixth-order ordinary differential equation requires six boundary conditions, including position, velocity, and acceleration constraints at the start and end points. This approach ensures high-quality trajectories for China EV applications.
To improve computational efficiency, the enhanced algorithm optimizes solution strategies and simplifies calculation processes. The computational complexity analysis is given by:
$$O(n \log n + k m^2)$$
where \(n\) is the number of trajectory discrete points, \(k\) is the number of iterations, and \(m\) is the number of constraints. By identifying computational bottlenecks, targeted optimizations reduce algorithm complexity. The algorithm employs a hierarchical optimization structure, decomposing the trajectory planning problem into dynamic weight optimization and smoothness optimization subproblems. Dynamic weight optimization uses a recursive approach based on an iterative update mechanism for weight factors. Specifically, at each planning cycle \(t\), the weight coefficients are updated using the formula:
$$w(t+1) = w(t) – \eta \nabla J[w(t)] + \mu (w(t) – w(t-1))$$
where \(w(t)\) is the weight vector at time \(t\), \(\eta\) is the learning rate (initialized to 0.2 and gradually decreased), \(\nabla J[w(t)]\) is the gradient of the objective function with respect to the weights, and \(\mu\) is the momentum factor (0.3) to accelerate convergence and avoid local oscillations. Initial weight values are set to predefined baseline weights and dynamically adjusted based on vehicle state and trajectory characteristics. The weight gradient is computed using finite difference methods: for each weight component \(w_i\), a small perturbation \(\Delta w\) is applied, and the change in the objective function is calculated to approximate the partial derivative \(\partial J / \partial w_i\). This efficient approach enables the algorithm to complete trajectory planning within 20 ms, representing a 31.2% improvement in computational efficiency, which is vital for real-time applications in electric vehicles.
For simulation validation and performance analysis, a MATLAB/Simulink-based simulation platform was developed, integrating vehicle dynamics models and lane change scenario modules. The simulation uses parameters from a typical China EV sedan, with adjustments for speed and road conditions to test algorithm adaptability under various scenarios. Key simulation parameters are summarized in Table 1.
| Parameter Name | Value | Unit |
|---|---|---|
| Vehicle Mass | 1850 | kg |
| Wheelbase | 2.78 | m |
| Front/Rear Track Width | 1.58/1.59 | m |
| Maximum Steering Angle | ±35 | ° |
| Steering Angular Velocity Limit | ±15 | °/s |
| Maximum Lateral Acceleration | 2.5 | m/s² |
| Lane Change Target Lateral Displacement | 3.75 | m |
| Motor Rated Power | 160 | kW |
| Motor Peak Torque | 310 | Nm |
| Battery Capacity | 70 | kWh |
| Simulation Step Size | 0.02 | s |
Based on the established simulation environment, the improved polynomial trajectory planning algorithm was compared with traditional fifth-order polynomial methods and B-spline curve methods. Under identical simulation scenarios and evaluation metrics, the algorithms were assessed for trajectory smoothness, computational efficiency, and control performance. Key performance indicators are compared in Table 2.
| Evaluation Metric | Improved Algorithm | Fifth-Order Polynomial | B-spline Curve |
|---|---|---|---|
| Maximum Lateral Acceleration (m/s²) | 2.1 | 2.8 | 2.5 |
| Maximum Curvature Change Rate (m⁻¹/s) | 0.042 | 0.068 | 0.055 |
| Trajectory Generation Time (ms) | 18.5 | 26.8 | 31.2 |
| Average Tracking Error (m) | 0.12 | 0.21 | 0.16 |
| Energy Consumption (kWh per maneuver) | 0.086 | 0.102 | 0.095 |
| Convergence Iteration Count | 8 | 12 | 15 |
The results demonstrate that the improved algorithm outperforms the comparison methods in key metrics such as trajectory smoothness, computational efficiency, and control accuracy, better meeting the practical needs of autonomous lane change control for electric vehicles. For example, trajectory smoothness improved by 23.5%, computational efficiency increased by 31.2%, and maximum lateral acceleration decreased by 18.7% compared to traditional fifth-order polynomial methods. These advancements highlight the algorithm’s potential for enhancing the performance of China EV in autonomous driving systems.
In conclusion, by constructing a new lane change trajectory planning algorithm based on an improved polynomial function, we have introduced a model that utilizes dynamic weight factors for fine-tuned trajectory characteristics. A multi-objective coordination optimization framework integrating trajectory smoothness and energy consumption characteristics was designed, incorporating a motor energy efficiency term in the objective function to reduce energy consumption by 15% during lane changes—a critical aspect for electric vehicles. The development of a trajectory smoothing optimization algorithm based on variational methods transformed the optimization problem into solving Euler equations, reducing the curvature change rate by 42% compared to traditional methods. An adaptive transition interval design method for trajectory segment junctions was proposed, with interval length dynamically adjusted based on speed, reducing trajectory continuity error by 78% in high-speed scenarios. Innovatively, a combination of soft constraint handling and adaptive step-size strategies was used for optimization solving, increasing the feasible solution rate by 18.7%. Through meticulous computational efficiency optimization, the algorithm achieves trajectory planning within 20 ms, improving computational efficiency by 31.2%. These contributions provide a superior lane change trajectory planning solution for the autonomous driving systems of electric vehicles, particularly in the context of China EV development, and pave the way for future research in real-time applications and integration with other autonomous driving modules.